where ,
are positive numbers. In this paper
we present some simple proofs of and several remarks (e.g.
historical) about the inequality and its corresponding continuous
version. Moreover, we discuss and comment on some very new
results. We also include some new proofs and results e.g. a weight
characterization of a general weighted Carleman type inequality
for the case 0 pq We also include some
facts about T. Carleman and his work.